Spin exchange optically pumped nuclear spin self compensation system for moving magnetoencephalography measurement

Abstract

Recording moving magnetoencephalograms (MEGs ), in which a person’s head can move freely as the brain’s magnetic field is recorded, has been a key subject in recent years. Here, we describe a method based on an optically pumped atomic co-magnetometer (OPACM) for recording moving MEGs. In the OPACM, hyper-polarized nuclear spins produce a magnetic field that blocks the background fluctuation low-frequency magnetic field noise while the rapidly changing MEG signal is recorded. In this study, the magnetic field compensation was studied theoretically, and we found that the compensation is closely related to several parameters such as the electron spin magnetic field, nuclear spin magnetic field, and holding magnetic field. Furthermore, the magnetic field compensation was optimized based on a theoretical model . We also experimentally studied the magnetic field compensation and measured the responses of the OPACM to different magnetic field frequencies. We show that the OPACM clearly suppresses low-frequency (under 1 Hz) magnetic fields. However, the OPACM responses to magnetic field frequencies around the band of the MEG. A magnetic field sensitivity of 3 fT/Hz^1/2 was achieved. Finally, we performed a simulation of the OPACM during utilization for moving MEG recording. For comparison, the traditional compensation system for moving MEG recording is based on a coil that is around 2 m in dimension , while our compensation system is only 2 mm in dimension .

1. Introduction

Magnetoencephalograms (MEGs ) are widely applied for the analysis and diagnosis of several diseases, including diagnosing early stages of Alzheimer’s disease [1], localizing the epileptogenic zone before epilepsy surgery [2], studying depression [3], and more. The spin-exchange-relaxation-free (SERF) atomic magnetometer [4], which can work under room temperature, has been developed; it competes with SQUID (superconducting quantum interference device) magnetometers, which are typical equipment for MEG recording. Shortly after its invention, the SERF atomic magnetometer began to be utilized for MEG study and the auditory evoked magnetic field was recorded [5–7].

Though there have been many studies about utilizing atomic magnetometers for acquiring MEGs, including studies examining the auditory evoked brain magnetic field [5,8], somatosensory evoked magnetic fields [9,10], visual evoked magnetic fields [11], etc., few studies have investigated the motor system while the subject is free to move [12]. The motor system is involved in the natural walking of a patient as the MEG is recorded, MEGs of the motor cortex recorded as the head moves, neurological disorder-induced essential tremors, MEGs of visual-motor integration, and so on.

Several studies have investigated moving MEGs. We know that the brain’s magnetic field is very small compared with the Earth’s magnetic field; thus, current SERF magnetometers must be used in a magnetically shielded room for MEG recording, and the residual magnetic field must be under 2 nT [13]. Both the residual magnetic fields in the shield in three directions and their gradients must be compensated. If there are magnetic field gradients, the magnetic field experienced by the sensors will change as the sensor is moving. Moreover, fluctuation from the environment, such as that caused by nearby subway trains, will also affect the magnetometers. The key to moving MEG recording is to develop an active compensation system to suppress the background magnetic field.

In the method developed by E. Boto et al. [12] for moving MEG recording, a 1.6 m×1.6 m bi-planar coil [13] is utilized to null the background magnetic field and its gradient. This wearable magnetometer system only can work in a $40cm\times 40cm\times 40cm$ space [12]. A feedback system was developed to maintain the residual magnetic field in this area below 2 nT in real time. However, the person’s head can only move slowly and must be kept within a small area as the MEGs are recorded. In the method developed by S. Mellor et al. [14], the spatial variation in the magnetic field is modeled and the model is used to predict the magnetic field resulting from movement-related artifacts [15,16]. The real-time brain magnetic field can be extracted by subtracting the artifact magnetic field from the magnetometer readings. Note that this method depends on predicting the magnetic field inside the magnetic field shielding room. Thus, it is only effective within a stable magnetic field. For a changing background magnetic field, such as the magnetic field originating from a subway train, its performance is expected to be worse. In the method developed by Pratt et al. [17], a whole-head 432-magnetometer optically-pumped MEG system is used to allow for comfortable head motion. In order to allow head motion, the system’s common mode rejection ratio can distinguish nearby signal sources from distant noise sources, such as the background fluctuation magnetic field noise. However, because this method utilizes a differential technique, only the gradient of the brain’s magnetic field can be measured. This approach cannot directly measure the brain’s magnetic field.

This article describes a new method to perform this type of background magnetic field compensation. Unlike the case of traditional atomic magnetometers, we use an optically pumped atomic co-magnetometer (OPACM) to perform moving MEG measurement. In the OPACM, nuclear spins fill the vapor cell, and the hyper-polarized nuclear spins produce a magnetic field that automatically shields the electron spins in the OPACM from the environment’s fluctuation magnetic field. This will leave only the electron spins that are sensitive to the higher-frequency MEG signal. This method does not need a large compensation coil system or a large feedback system. Our compensation system works in situ and is only approximately $2mm\times 2mm\times 2mm$ in size. This would allow a person to move more naturally in the magnetic field shielding room and enable us to record MEGs as the person is walking. Moreover, our method records MEGs directly without the differential of the background noise. Our method is also effective in contexts with background magnetic field noise.

The automatically compensated process is similar to the self-compensation effect in a SERF co-magnetometer [18,19]. Because of their self-compensating ability for the DC magnetic field, SERF atomic co-magnetometers have been utilized for rotation sensing [20,21] and studying physics beyond the standard model [22–24]. For rotation sensing, the atomic spins are directed in the inertial space and their direction can easily be disturbed by background magnetic field fluctuation. Thus, the self-compensating effect could greatly reduce the co-magnetometer’s sensitivity to magnetic fields. For physics applications beyond the standard model, a similar situation occurs.

Note that atomic magnetometers fabricated by micro-machining technology are also rapidly developing; in such cases, the size of the magnetometer is greatly reduced. This is a fascinating research trend, including some good studies such as [25,26]. Atomic magnetometers can record not only MEGs, but also fetal magnetocardiography measurements [27]. We believe that our OPACM could also be fabricated by this type of micro-machining technology, and we are now developing OPACMs using Micro-Electro-Mechanical System(MEMS) technology.

The paper is organized as follows. The second section discusses the theoretical basis for our study. Section 3 describes our experimental setup. Section 4 shows our results, and Section 5 provides a discussion of these findings. Finally, Section 6 concludes this paper.

2. Theory

As shown in Fig. 1 , owing to the ultra-low intensity of the brain’s magnetic field, MEG measurements are performed in a magnetic field shielding room with an inner space that is typically approximately $2m\times 2m\times 2m$ in size [28]. Further magnetic field compensation is carried out by the coils inside the room. The residual magnetic field around the head area, which is typically $0.5m\times 0.5m\times 0.5m$ in size [13], can generally be reduced to under 2 nT with active compensation of the background magnetic field and its gradient by the coils. However, this magnitude is still much larger than that of the brain magnetic field. The gradient of the residual magnetic field will cause substantial variation as the head moves in the room. The typical SERF atomic magnetometer is sensitive to both the background fluctuation magnetic field and the brain magnetic field. We notice that the background magnetic field and its fluctuation are typically slowly changing magnetic fields , while the brain magnetic field changes rapidly. If we can automatically compensate the background low-frequency magnetic field and leave the magnetometer sensitive to the rapidly changing brain magnetic field, the brain’s magnetic field can be recorded as the head moves in the magnetic shielding room.

Fig. 1.

This image illustrates brain magnetic field measurement as the head is moving. The subject of MEG recording stays in a magnetic field shielding room. Further magnetic field compensation is performed by the planar coil. As the head moves, the sensors typically experience a fluctuation magnetic field of approximately 1 nT due to the inhomogeneity of the background field. This field strength is much larger than that of the brain magnetic field. Thus, we have developed an automatic compensation system using hyper-polarized nuclear spins. This nuclear spin shielding allows the OPACM to remain sensitive to the rapidly changing brain magnetic field. The brain magnetic field is recorded by the sensors, and the control system analyzes the signal.

The key to measuring the brain magnetic field is the SERF magnetometer. The very high sensitivity of the SERF magnetometer is a result of reducing the spin exchange relaxation between electron spin collisions. The idea of spin exchange relaxation suppression was first developed by William Happer et al. [29]. The SERF atomic magnetometer was not invented until 2002 [30], but its sensitivity surpasses that of a SQUID magnetometer [4]. In a SERF atomic magnetometer, the electron spins of the alkali atoms are optically pumped by a laser, and the spins rotate if they experience a magnetic field. Either the optical rotation or the laser absorption method could be utilized to detect the polarized spins’ direction. Thus, the magnetic field could be indirectly deduced using the optical detection method.

If we fill nuclear spin $I$ in the vapor cell, under certain conditions, these nuclear spins will shield the electron spins from the outside fluctuation magnetic field. As shown in Fig. 2, nuclear spins with momentum $I$ and electron spins with momentum $S$ fill the vapor cell. The electron spins $S$ are optically pumped by the laser light and point in the direction of the laser light . Then, the nuclear spins $I$ become hyper-polarized by the electron spins and their polarization direction becomes the same as that of the electron spins [31]. Owing to the Fermi contact interaction [32], the hyper-polarized nuclear spins produce a magnetic field that could affect the electron spins. In a spherical vapor cell, the effective magnetic field will be approximately $B^n=8\pi k_0{\mu }_n[N]P^n/3$ [33]. In this equation, $B_n$ is the magnetic field produced by the nuclear spins, such as ${}^{21}Ne$ or ${}^{31}Xe$ . $k_0$ is an enhancement factor [32] that could enhance the magnetic field experienced by the electron spins during spin exchange collision. ${\mu }_n$ is the nuclear magnetic moment for nuclear spin $I$ . $[N]$ is the number density of the nuclear spins and $P^n$ is the polarization of the nuclear spins. Note that the ${}^{21}Ne$ nuclear magnetic moment $\mu ({}^{21}Ne)$ is 0.66 ${\mu }_N$ , in which ${\mu }_N$ is the nuclear magnetic moment of the neutron and the ${}^{31}Xe$ nuclear magnetic moment $\mu ({}^{31}Xe)$ is 0.69 ${\mu }_N$ . As with the nuclear spins, the electron spins also produce a magnetic field $B_e$ , which affects the nuclear spins and $B_e$ is equal to $8\pi k_0{\mu }_B[A]P^e/3$ . Here, ${\mu }_B$ is the Bohr magneton, $[A]$ is the number density of the electron spins, and $P^e$ is the polarization of the electron spins. Under steady-state conditions, the polarizations of the nuclear spins and electron spins, as well as $B_n$ and $B_e$ , will be stable. In order to realize the automatic compensation effect, we add a compensation magnetic field $B_c$ in the opposite direction of $B_n$ ; the strength of $B_c$ is equal to the strength of $B_n$ + $B_e$ .

Fig. 2.

The principle of the fluctuating background magnetic field compensation.

As shown in Fig. 2, if there is a disturbance from a background magnetic field $B_x$ , the total magnetic field ${\mathbf{B}}_{\mathbf{t}\mathbf{o}\mathbf{t}}$ experienced by the nuclear spins will be $\mathbf{(}{\mathbf{B}}_{\mathbf{c}}\mathbf{-}{\mathbf{B}}_{\mathbf{e}}\mathbf{)}\mathbf{+}{\mathbf{B}}_{\mathbf{x}}$ . Since $B_c-B_e$ is equal to $B_n$ , under a small disturbance from the background magnetic field (meaning that $B_x$ is much smaller than $B_n$ ), the projection of ${\mathbf{B}}_{\mathbf{n}}$ along the $x$ direction $B_n^x$ is equal to $B_x$ , and they are in opposite directions. Thus, the electron spins will still stay in the pumping laser direction. We say that the nuclear spins automatically compensate the fluctuation magnetic field and shield the electron spins from the disturbing magnetic field.

Note that in theory, the nuclear spins can perfectly shield the DC disturbing magnetic field, while the AC magnetic field from the brain cannot be compensated perfectly by the nuclear spins because of the finite bandwidth of the nuclear spins’ response to the magnetic field. This will allow a magnetometer that is only sensitive to the rapidly changing brain magnetic field to automatically compensate the background low-frequency disturbing magnetic field . To further study this mechanism in detail, we need to solve the Bloch equations of this system and study the response of the OPACM to several frequencies of magnetic fields.

In this paper, we mainly focus on an OPACM based on K-Rb- ${}^{21}\mathrm{N}\mathrm{e}$ . The electron spins are mainly from the Rb atoms. Because K atoms are utilized as the hybrid optical pumping atoms, the K atom spins typically account for only a very small part of the electron spins. Most of the electron spins are from the Rb atoms [34]. Thus, we can neglect the electron spins from the K atoms. The full Bloch equations can be simplified to include only two spin species. There have been various works describing the full Bloch equations for K-Rb- ${}^{21}\mathrm{N}\mathrm{e}$ OPACM [19,35]. Here, we briefly list them.

$$\begin{array}{c}\frac{\mathrm{\partial }{\mathbf{P}}^{\mathbf{e}}}{\mathrm{\partial }t}=\frac{{\gamma }^e}{Q(P^e)}(\mathbf{B}+{\lambda }_{Rb-Ne}{M}_0^n{\mathbf{P}}^{\mathbf{n}})\times {\mathbf{P}}^{\mathbf{e}}-\\ \frac{1}{T_{2e},T_{2e},T_{1e}}{\mathbf{P}}^{\mathbf{e}}/Q(P^e)+(R_p{\mathbf{s}}_{\mathbf{p}}-R_p{\mathbf{P}}^{\mathbf{e}})/Q(P^e)\end{array}$$ (1)

$$\begin{array}{c}\frac{\mathrm{\partial }{\mathbf{P}}^{\mathbf{n}}}{\mathrm{\partial }t}={\gamma }^n(\mathbf{\Omega }/{\gamma }^n+\mathbf{B}+{\lambda }_{Rb-Ne}{M}_0^e{\mathbf{P}}^{\mathbf{e}})\times {\mathbf{P}}^{\mathbf{n}}+\\ R_{Rb-Ne}^{se}({\mathbf{P}}^{\mathbf{e}}-{\mathbf{P}}^{\mathbf{n}})-\frac{1}{T_{2n},T_{2n},T_{1n}}{\mathbf{P}}^{\mathbf{n}}\end{array}$$ (2)

Here, ${\mathbf{P}}^{\mathbf{e}}$ and ${\mathbf{P}}^{\mathbf{n}}$ are the Rb electron and ${}^{21}\mathrm{N}\mathrm{e}$ nuclear spin polarizations, $\mathbf{\Omega }=\{{\mathrm{\Omega }}_x,{\mathrm{\Omega }}_y,{\mathrm{\Omega }}_z\}$ is the rotation angular velocity input, $\mathbf{B}$ is the external magnetic field including the compensation field ${\mathbf{B}}_{\mathbf{c}}$ and the disturbing field ${\mathbf{B}}_{\mathbf{x}}$ , $M_0^e={\mu }_B[A]$ and $M_0^n={\mu }_N[N]$ are the magnetizations of the electron and nuclear spins as the spins are fully polarized, ${\lambda }_{Rb-Ne}$ is equal to $8\pi k_0/3$ for a spherical alkali vapor cell, $R_p$ is an effective pumping rate that is related to the K pumping rate and the density ratio of K to Rb [34], ${\mathbf{s}}_{\mathbf{p}}$ is the optical pumping vector along the propagation of the pump with a magnitude equal to the degree of circular polarization, and $R_{Rb-Ne}^{se}$ is the spin exchange rate of ${}^{21}\mathrm{N}\mathrm{e}$ nuclear spins. The precession frequency of the alkali metal is slowed by the factor $Q(P^e)$ [36], which is related to the alkali spin polarization. $T_1$ and $T_2$ are the relaxation times for components of the polarization parallel and transverse to ${\mathbf{B}}_{\mathbf{z}}$ , respectively. The subscripts $e$ and $n$ denote electron spins and nuclear spins, respectively.

Here, we are interested in the OPACM’s sensitivity to external fluctuation magnetic fields. Thus, we need to study the OPACM’s responses to external magnetic field under different frequencies. A sinusoidal signal will be applied to the Bloch equations to obtain the output amplitude.

It is reasonable to assume that the disturbing field is much smaller than the compensation field; thus, the polarizations of the electron spin and the nuclear spin in the $z$ direction are constant. We can discard the $P_z^e$ and $P_z^n$ terms in Eq. (1) and Eq. (2). We define $\mathbf{P}={\{P_x^e,P_y^e,P_x^n,P_y^n\}}^T$ , and Equations (1) and (2) can be simplified to be $d\mathbf{P}/dt=\mathbf{A}\bullet \mathbf{P}+\mathbf{C}$ . Nearly 5 hours are needed to polarize ${}^{21}\mathrm{N}\mathrm{e}$ ; thus, it is reasonable to assume that $R_{Rb-Ne}^{se}$ and the relaxation rate of ${}^{21}\mathrm{N}\mathrm{e}$ are small numbers. The equations can be simplified, and $\mathbf{A}$ is equal to:

$$\left(\begin{array}{cccc}\widetilde{-R_{tot}^e}& -{\omega }_e& 0& {\omega }_{en}\\ {\omega }_e& \widetilde{-R_{tot}^e}& -{\omega }_{en}& 0\\ 0& {\omega }_{ne}& 0& {\omega }_n\\ -{\omega }_{ne}& 0& {\omega }_n& 0\end{array}\right)$$ (3)

In the above equation, $\widetilde{-R_{tot}^e}$ is equal to $(1/T_{2e}+R_p)/Q(P^e)$ , ${\omega }_{en}$ is equal to ${\gamma }^e{\lambda }_{Rb-Ne}{M}_0^n{P}_z^e/Q(P^e)$ , ${\omega }_{ne}$ is equal to ${\gamma }^n{\lambda }_{Rb-Ne}{M}_0^e{P}_z^n$ , and ${\omega }_n$ is equal to ${\gamma }^n(B_c-B_e)$ . The input term $\mathbf{C}$ is equal to:

$$\begin{array}{c}\{\widetilde{b_y^e}=P_z^e{\gamma }^e{B}_y/Q(P^e),\widetilde{-b_x^e}=-P_z^e{\gamma }^e{B}_x/Q(P^e),\\ {\widetilde{b_y^n}=P_z^n{\gamma }^n{B}_y,\widetilde{-b_x^n}=-P_z^n{\gamma }^n{B}_x\}}^T\end{array}$$ (4)

Suppose that there are two oscillating magnetic field inputs in the $x$ and $y$ directions:

$$\mathbf{B}=(B_{0x}\mathbf{x}+B_{0y}\mathbf{y})e^{-i\omega t}$$ (5)

Under steady-state conditions, the polarization of the electron spin and the nuclear spin can be solved:

$$\mathbf{P}={(\mathbf{A}-i\omega \mathbf{I})}^{-1}\bullet \mathbf{C}$$ (6)

In the experiment, the $x$ polarization of the electron spins is detected using a probe light. Under typical conditions, the electron precession frequency ${\omega }_e$ is much larger than the nuclear spin frequency ${\omega }_n$ and the input magnetic field frequency $\omega$ . Equation (6) can be further simplified as:

$$P_x^e\approx \frac{B_{0x}{\gamma }^e{P}_z^e\omega (\omega {\omega }_e-i\widetilde{R_{tot}^e}{\omega }_n)e^{-i\omega t}/Q(P^e)}{[\widetilde{R_{tot}^e}(\omega -{\omega }_n)+i\omega {\omega }_e][\widetilde{R_{tot}^e}(\omega +{\omega }_n)-i\omega {\omega }_e]}$$ (7)

Equation (7) contains both the real part and the image part. The real part represents the amplitude of the output signal, while the image part represents the phase shift. Thus, we only consider the real part to deduce the response of the OPACM to the external magnetic field. Note that Equation (7_ only considers the $x$ oscillating magnetic field response. If we consider both the $x$ and the $y$ directions, the equation can be further simplified:

$$\begin{array}{c}{P}_x^e\approx P_z^e\frac{{\gamma }^e{B}_{0x}\omega e^{-i(\omega t+{\mathrm{\Phi }}_x)}}{R_{tot}^e{[{\omega }_n^2+{\omega }^2{\omega }_e^2/{(R_{tot}^e/{\gamma }^e)}^2]}^{1/2}}\\ +P_z^e\frac{{\gamma }^e{B}_{0y}{\omega }^2{e}^{-i(\omega t+{\mathrm{\Phi }}_y)}}{R_{tot}^e[{\omega }_n^2+{\omega }^2{\omega }_e^2/{(R_{tot}^e/{\gamma }^e)}^2]}\end{array}$$ (8)

${\mathrm{\Phi }}_x$ and ${\mathrm{\Phi }}_y$ are the phase shifts between the input oscillating magnetic field and output signal for the $x$ and $y$ directions, respectively. From Eq. (8) we can see that the output signal is frequency-dependent and related to ${\omega }_n$ , ${\omega }_e$ , and $R_{tot}^e$ . Under a low frequency, the output signal is proportional to $\omega$ in the $x$ direction and ${\omega }^2$ in the $y$ direction. The lower the frequency, the smaller the output signal. This means that the OPACM could efficiently suppress the low-frequency disturbing background magnetic field while remaining sensitive to the higher-frequency brain magnetic field. We can also change the parameters to let the OPACM work in a proper frequency range. Note that numerical solution could also be obtained based on Equations (1) and (2). The analytical solution in Eq. (8) has a certain limitation; the solution assumes that the input magnetic field frequency $\omega$ is much smaller than ${\omega }_e$ . At high frequencies, the result will deviate from the solution. In this case, the numerical solution should be utilized.

3. Experimental setup

The configuration of the OPACM is similar to the configurations described in the references [19,37]. A schematic of the experimental setup is shown in Fig. 3. The homemade vapor cell utilized in this experiment was spherical and the material was aluminosilicate. The cell’s diameter was 14 mm and it contained a small droplet of K-Rb ( ${}^{85}Rb$ (72.2 $\mathrm{\%}$ ) and ${}^{87}\mathrm{R}\mathrm{b}$ (27.8 $\mathrm{\%}$ )) mixture. The cell was also filled with ${}^{21}\mathrm{N}\mathrm{e}$ gas (70 $\mathrm{\%}$ isotope enriched) under a pressure of about 3 atm and ${\mathrm{N}}_2$ gas under about 40 Torr for quenching . The mole fraction ratio of K in the mixture was approximately 0.05. A homemade 110 kHz AC electrical heater was used to non-magnetically heat the vapor cell. For better thermal conductivity, a nitride ceramic post was connected between the vapor cell and the heater. The heater and vapor cell were kept in a vacuum chamber made of PEEK. A 0.1 Pa vacuum was achieved by using 10 L/s turbo molecular pumps for thermal insulation. Several layers of $\mu$ -metal magnetic field shields together with a layer of 10-mm-thick ferrite [38] were used to shield the vapor cell from Earth’s magnetic field and other fluctuation magnetic fields. The diameter of the ferrite was 100 mm. Coils were utilized for further residual magnetic field compensation. To achieve high ${}^{21}\mathrm{N}\mathrm{e}$ polarization, the K-Rb hybrid pumping technique [39] was utilized [40]. K atoms were directly pumped and Rb atoms were polarized through spin exchange optical pumping with K atoms. Because of the low efficiency of the hybrid optical pumping, most of the laser light was wasted, and a laser with a power of approximately 1W power was utilized. To efficiently pump the atoms, the pumping light was reflected by a mirror again after passing through the vapor cell. The projection of Rb electron spin along the probe light direction $P_x^e$ was detected by a linearly polarized light. If there was polarization projection along the probe light direction, the initial polarization plane $P_1$ would rotate after passing the vapor cell and the final polarization plane would be $P_2$ . The polarization rotation was detected via the photoelastic modulation method [19]. The probe distributed feedback (DFB) laser was detuned approximately 0.4 nm away from the absorption center of the Rb D1 line.

Fig. 3.

The schematic of the experimental setup.

4. Results

The key to a good simulation is to determine the appropriate parameters for the simulation. Here, we list those used for our simulation. The temperature of the vapor cell was around 473 K, which was measured using a temperature sensor. From the mole fraction ratio, we calculate that the number densities of K and Rb are $n_K=8.1\times {10}^{12}/c{m}^3$ and $n_{Rb}=8.7\times {10}^{14}/c{m}^3$ respectively. The number density of Rb is much larger than that of K. Because $n_{Rb}$ is very large, the electron spin produces a magnetic field of approximately $B^e$ =100 nT, which is experienced by the ${}^{21}\mathrm{N}\mathrm{e}$ nuclear spins. This magnetic field was acquired as the Rb polarization was around 50%, which is the optimized polarization. Under the 50% polarization condition, the OPACM demonstrates the best response. Furthermore, the nuclear spins produces a magnetic field of approximately $B^n$ =500 nT. The total relaxation of Rb includes the optical pumping rate, the Rb-Rb spin destruction rate, the Rb-Ne spin destruction rate, and the spin exchange relaxation. Particularly in the case of the K-Rb- ${}^{21}\mathrm{N}\mathrm{e}$ OPACM, the large $B^e$ value results in a large magnetic field experienced by the Rb electron spins, which leads to the spin exchange relaxation of Rb [19].

Based on the method described in the reference [19], the optical pumping rate of Rb $R_p$ is 1950 ${\mathrm{s}}^{-1}$ and the total relaxation rate of Rb $R_{tot}^e$ is 3900 ${\mathrm{s}}^{-1}$ . The pump laser power density is 640 ${\mathrm{mW/cm}}^2$ . Under this power, the Rb polarization is around 50%. We also measured the spin destruction rate of Rb to be ${\mathrm{480s}}^{-1}$ . We subtracted the optical pumping rate and spin destruction rate from the total relaxation rate, thus determining the spin exchange relaxation of Rb-Rb to be 1470 ${\mathrm{s}}^{-1}$ . Under the temperature used in our experiment, the spin exchange rate between K and Rb is $1.6\times {10}^5{s}^{-1}$ . This rapid spin exchange rate mixes K and Rb together, causing them to behave like a single species [34]. K atoms are directly pumped by the laser. Through the optical pumping rate of Rb and the number density ratio of Rb to K, we can calculate that the optical pumping rate for K is 210000 ${\mathrm{s}}^{-1}$ .

In order to acquire the responses of the OPACM to different frequencies of magnetic field, we theoretically applied the oscillating magnetic field to Eqs. (1) and (2). With the experimental parameters, we can calculate the output response $P_x^e$ . The input magnetic field amplitudes in the $x$ and $y$ directions are the same: $B_{0x}=B_{0y}=0.08nT$ . This magnetic field is much smaller than the equivalent magnetic field line width. Thus, the OPACM works in the linear area. As the OPACM probes the $x$ polarization of the alkali metal spins, we directly provide the amplitude of the $x$ polarization sinusoidal output signal. Figure 4 shows both the theoretical and experimental results of the $x$ and $y$ magnetic field responses. The simulation results were determined through the numerical calculation based on Eqs. (1) and (2), and we fit the experimental results to the simulation results. In Fig. 4, “ $B_x$ theory" and “ $B_y$ theory" denote the theoretical simulation of the response of the OPACM to the oscillating magnetic fields under different frequencies. The frequency range is from 0.03 Hz to 200 Hz. “ $B_x$ Experiment" and “ $B_y$ Experiment" denotes the experimental results. In the experiment, we applied both the $x$ and $y$ direction magnetic fields to the OPACM, then recorded the output signal amplitude. The output signal’s frequency is the same as that of the input signal. The input magnetic field amplitude is 0.08 nT. Figure 4 shows that the experimental and theoretical results fit well with each other. This result provides strong evidence that our simulation model is correct and the parameters in the model are reasonable.

Fig. 4.

The responses of the OPACM to several input signals.

It is not surprising that the frequency responses of the OPACM to magnetic fields at frequencies under 1 Hz are small. We have theoretically shown that the OPACM is not sensitive to DC magnetic fields owing to the nuclear spins’ self-compensation mechanism. From Eq. (8), we can simplify the equations under very low $\omega$ values. We determine that the amplitude of the $x$ magnetic field response is proportional to $\omega /{\omega }_n$ . As the frequency of the magnetic field is reduced to 1/10 of the original frequency, the amplitude is also reduced to 1/10 of the original amplitude. For the $y$ direction, the amplitude is proportional to ${(\omega /{\omega }_n)}^2$ . This means that if the frequency of the magnetic field is reduced to 1/10 of the original frequency, the amplitude is reduced to 1/100 of the original amplitude. The OPACM can suppress magnetic fields more effectively in the $y$ direction. At higher frequencies, the response is larger. This is because of the fact the nuclear spins and the electron spin ensembles are coupled together at low frequencies to automatically compensate the input magnetic field. At low frequencies, the nuclear spin could fast enough to response to the outside magnetic field change. Thus, the low frequency magnetic field fluctuation could be compensated. At high frequencies, due to the small nuclear spin magnetic moment, the nuclear could hardly response to the external magnetic field changing. However, the electron spin owns a very large magnetic moment which is around 1000 times larger than that of the nuclear spin. Thus, the electron spin could easily responses to the external magnetic field changing. That is the reason why the OPACM has the ability to do MEGs measurement.

Because the OPACM is sensitive to rotation [19,20], we also theoretically studied the OPACM’s responses to the $x$ and $y$ angular velocity input. Like the magnetic field, oscillating ${\mathrm{\Omega }}_x$ and ${\mathrm{\Omega }}_y$ values were applied to the OPACM and the output amplitudes were recorded under different frequencies. From Eq. (2), we can see that the rotations are equivalent to the magnetic field divided by the gyroscopic ratio ${\gamma }^n$ . Thus, we set the rotation angular velocity amplitudes to 0.08 nT, which is the same as the magnetic field input amplitude. Figure 4 shows that under low frequencies, the OPACM is sensitive to the $y$ angular velocities. If the input angular velocity frequency is much larger than 1 Hz, the output amplitude signal is greatly suppressed. In contrast, for ${\mathrm{\Omega }}_x$ , the OPACM is not sensitive to the $x$ rotations. Note that out experiment was performed on a large and heavy platform, which cannot be driven by a rotation platform . Thus, we only show the theoretical results of the rotation angular velocity responses. This subject could be studied in the future using a miniature OPACM.

In order to determine whether the OPACM could suppress a slowly changing magnetic field while retaining its sensitivity to the rapidly changing brain magnetic field, we performed a theoretical response study in the times domain . As shown in Fig. 5, the input magnetic fields in the $x$ and $y$ directions are the same. The signal includes a rapidly changing pulse signal and a slowly changing sinusoidal signal. The frequency of the low-frequency sinusoidal signal is 0.1 Hz, which fulfills the requirement of low-frequency conditions from Fig. 4. For the brain magnetic field, we assume that the width of the pulse is under 0.25 s, which is reasonable, as it has been reported that the width of the auditory evoked brain magnetic field is around 0.25 s [5]. The dashed lines show the responses of the OPACM. For a better comparison, we set the peak of the pulse to be the same as the peak of the low frequency sinusoidal signal. As shown, there is a clear peak and low frequency magnetic field suppressed output signal. For the $x$ magnetic field response, the peak of the rapidly changing pulse response is 0.001, which is 4 times larger than that of the low frequency sinusoidal input. In contrast, for the $y$ magnetic field response, the pulse peak output is very small, and the slowly changing sinusoidal magnetic field is nearly totally suppressed. Even though the $y$ direction can suppress low-frequency magnetic fields more effectively, the response of the brain magnetic field is also greatly suppressed. Thus, we can use the $x$ channel response to probe the brain magnetic field.

Fig. 5.

The OPACM’s responses to a fast pulse magnetic field and a slowly changing sinusoidal magnetic field in both of the $x$ and $y$ directions.

We also measured the sensitivity of the OPACM and Fig. 6 shows the result. We measured the sensitivity using the typical noise spectral method [19]. We let the OPACM stay still and zeroed the residual magnetic fields in all three directions. The output signal, that is, the voltage signal, was recorded by a data acquisition system. We needed to change the output signal into a magnetic field signal through the scale factor. The scale factor was measured based on the method given in [19,37]. The power spectral density was calculated based on the time zone magnetic field signal. In order to clearly identify the noise floor, we averaged the power spectral density in very 0.1 Hz . That is, in every 0.1 Hz bin, the average noise amplitude represents the noise in the frequency range. Note that the frequency responses of the OPACM to different frequency magnetic field vary. Thus, the scale factors are also frequency dependent. For example, the OPACM is not sensitive to low-frequency noise; thus, the scale factors in the low-frequency area are small, whereas at larger frequencies, the scale factors are larger. The scale factors for the $x$ and $y$ directions are also different. The method for acquiring the sensitivity of the OPACM is similar to the method described in the references [4,41]. It is evident that our OPACM is less sensitive to the magnetic field in the $y$ direction. Thus, the $y$ direction has worse sensitivity. In Fig. 6, the solid line shows the magnetic field sensitivity in the $x$ direction. There are several noise sources that could affect the sensitivity of the OPACM. We observed peaks in the frequency range of 1-5 Hz; we determined that this noise came from the vibration of our optical board, which is supported by a mechanical vibration isolation system, which can cause rotational vibration in that frequency range. We have noted that our OPACM is sensitive to angular velocity. At lower frequency ranges, the 1/f noise dominates the noise band. At the frequency range of 5-10 Hz, a sensitivity of 3.2 ${\mathrm{fT/Hz}}^{1/2}$ was achieved. For frequencies larger than 20 Hz, the noise source clearly decreases with increasing frequency. This is because we used a low pass filter with a 3 dB bandwidth of 20 Hz to limit our OPACM to 20 Hz. This was done because our OPACM has a limited bandwidth and the auditory evoked brain magnetic field is also in this frequency range. The sensitivity of our magnetometer is also limited by the responses of the magnetometer. The response is low compared with the equivalent rotation input. This is because that the nuclear spins are coupled with the electron spins. Even at frequency above 1Hz, the magnetic field signal is also attenuated by the nuclear spin because of the nuclear self compensation effect. We need to let the resonance frequency of the nuclear spin be low enough and thus the responses of the OPACM could be better at the frequency range of 1-200Hz. Moreover, the sensitivity of our magnetometer is limited by the pumping laser power noise. We need to improve the relaxation of the electron spins and thus the responses of the magnetometer could be larger. However, there seems to be little method could achieve this.

Fig. 6.

The sensitivity of the OPACM.

For comparison, we also show a dotted line to illustrate the sensitivity of the OPACM if there is no low frequency magnetic field noise compensation in the $x$ direction. In such a case , the noise power is clearly larger at frequencies smaller than 1 Hz. This clearly show that the low-frequency magnetic field noise compensation effect works. We also measured the sensitivity of the OPACM in the $y$ direction. As previously shown, the OPACM is less sensitive to magnetic field in the $y$ direction. In noise spectra under 1 Hz, the $y$ direction noise still goes down while that of the $x$ direction is nearly flat. Under low frequencies, the noise in the $y$ direction is smaller than that of the $x$ direction. This means that the $y$ direction has a better magnetic field compensation effect.

It has been reported that the magnetic field noise from the magnetic field shielding material could dominate the noise source [38,42]. Thus, we also calculated our ferrite magnetic field noise. Based on the theory [38], the thermal noise of the domain in the ferrite could produce low-frequency magnetic field noise. At higher frequencies, the random movement of the electrons in the ferrite could also produce magnetic field noise. In Fig. 6, the dash-dotted line shows the magnetic field noise originating from the ferrite, which has an inner diameter of 10 cm, a thickness of 1 cm, and a length of 30 cm. This noise level is clearly smaller than the OPACM’s sensitivity.

5. Discussion

Rotation could affect the sensitivity of the OPACM. This is because the nuclear spins very clearly respond to rotation . Thus, if we want to record moving MEGs, we need to avoid rotational movement. Figure 5 only shows a simulation of the OPACM. We are currently attempting to build a magnetic field shielding room for real measurement of moving MEGs. This paper only provides the magnetic field compensation ability and the sensitivity of the OPACM. We also believe that better magnetic field compensation could be achieved by optimizing parameters such as the magnetic fields of the electron spins and nuclear spins. The nuclear spins utilized in this paper are ${}^{21}\mathrm{N}\mathrm{e}$ . A period of five hours is necessary to polarize their spins. This time is so long that it could limit the robustness of the OPACM. Thus, other nuclear spins such as ${}^{129}\mathrm{X}\mathrm{e}$ and ${}^{131}\mathrm{X}\mathrm{e}$ should be studied. For comparison, the method utilized in the reference [12], which aims to perform moving MEG measurement, is based on coil system compensation. It is clear that magnetic field gradients could affect the compensation because only one magnetometer is utilized as the feedback of coil compensation. Thus, a complicated gradient compensation system must be utilized; furthermore, the measurement area is only $40cm\times 40cm\times 40cm$ . However, in our OPACM, the nuclear spins in each OPACM could compensate the background magnetic field automatically and provide in-situ magnetic field compensation. The magnetic field gradients do not need to be compensated, and our measurement area will be larger.

Traditionally, the OPACM is utilized for rotation measurement and it is utilized as a gyroscope. As we know, the drift of a gyroscope is a very important problem since it could decide the performance of a gyroscope [43]. Magnetic field fluctuation could be a main gyroscope drift source since it directly affects the spin direction. The nuclear self- compensation could effectively suppress the magnetic field drift in an atomic spin gyroscope since the nuclear spins could automatically compensate the magnetic field fluctuation. We believe that low frequency magnetic field fluctuation could also be a problem in an atomic magnetometer for MEGs recording. The current method for the fluctuation field suppression is based on a coil compensation system or magnetic field differential method which have been talked in the Introduction section. In our OPACM, the drift of the magnetic field was compensated by the nuclear spin which is similar to the atomic spin gyroscope. The difference is that in MEGs recording, higher frequencies are concerned while in the atomic spin gyroscope, the drift falls in the low frequency range. The nuclear spin self-compensation method is both good for the gyroscopes and the MEGs recording sensors.

The miniaturization of the OPACM is very important for future application. The current OPACM is very large and it was performed on an optical table. The core of the magnetometer is the vapor cell which is 14mm in diameter. This vapor cell is still too large for future sensor head development. In the future, a single beam OPACM could be developed which is similar to the one described in this paper [44]. The pumping laser and the probe laser could be combined to be one single beam and laser absorption signal could be detected. Even though the sensitivity could be worse(an order of lower in the sensitivity), the complicate photo-elastic modulation system could be eliminated in the single beam configuration. In the single beam configuration, the spherical vapor cell could be changed into a cylinder vapor cell whose diameter could be as small as 3mm and the height could be 3mm. Moreover, combined with the micro-fabricating method, the OPACM could be miniaturized even smaller. Thus, it is reasonable to develop very small sensor head in the future.

In the current experiment, ${}^{21}\mathrm{N}\mathrm{e}$ isotope enriched nuclear spins are utilized. Unfortunately, the nature abundance of ${}^{21}\mathrm{N}\mathrm{e}$ is very low and thus it is very expensive to acquire ${}^{21}\mathrm{N}\mathrm{e}$ gases. Moreover, due to the quadru-polar interaction induced relaxation of ${}^{21}\mathrm{N}\mathrm{e}$ - ${}^{21}\mathrm{N}\mathrm{e}$ collision, it is hard to hyper-polarize the nuclear spins. Temperature around 473K was utilized and the aluminosilicate glass material was utilized to fabricate the vapor cell. Hybrid optical pumping with K-Rb mixture was used. Other isotope enriched noble gases such as ${}^{129}\mathrm{X}\mathrm{e}$ , ${}^{131}\mathrm{X}\mathrm{e}$ , ${}^3\mathrm{H}\mathrm{e}$ could also be utilized. For example, we can use the Rb- ${}^{129}\mathrm{X}\mathrm{e}$ pair. The limitation is that the sensitivity could be obvious lower due to the large spin destruction cross section between Rb and ${}^{129}\mathrm{X}\mathrm{e}$ . We can benefit from the easier hyper-polarization of the Xe nuclear spins. The ${}^{131}\mathrm{X}\mathrm{e}$ nuclear spins own the similar problem. Take ${}^3\mathrm{H}\mathrm{e}$ for another example, we could choose Rb- ${}^3\mathrm{H}\mathrm{e}$ pair. The sensitivity could be obvious higher due to the small spin destruction cross section between Rb and ${}^3\mathrm{H}\mathrm{e}$ . However, ${}^3\mathrm{H}\mathrm{e}$ is quite hard to be polarized and vapor cells with very high pressure of ${}^3\mathrm{H}\mathrm{e}$ gas was compulsory. This could make the alkali vapor cell fabrication process hard. The low polarization of ${}^3\mathrm{H}\mathrm{e}$ could also limit the magnetic field compensation range of the OPACM.

6. Conclusion

In conclusion, we have developed an OPACM that automatically compensates the background fluctuation magnetic field in situ. In the OPACM, isotope enriched ${}^{21}\mathrm{N}\mathrm{e}$ atoms are hyper-polarized through spin exchange optical pumping. The polarized nuclear spins will automatically trace the direction of the outside fluctuating magnetic field and produce a magnetic field in the opposite direction, which will cancel the fluctuating magnetic field. This will shield the alkali atoms’ spin, which is utilized for brain magnetic field sensing based on the fluctuating magnetic field. Owing to the bandwidth of the shielding effect, the higher-frequency brain magnetic field will be sensed by the OPACM while the low-frequency fluctuating background magnetic field will be cancelled.

We have theoretically shown the shielding effect and studied its relationship with $B^e$ and $B^n$ . We have also experimentally studied the compensation effect. The simulation of the OPACM’s response to the brain magnetic field, as well as the slowly changing background magnetic field, was also studied. Finally, we measured the sensitivity of the OPACM, determining that a sensitivity of 3.2 ${\mathrm{fT/Hz}}^{1/2}$ in the $x$ direction was achieved.

Funding

China Postdoctoral Science Foundation10.13039/501100002858 (2020M683462); Natural Science Foundation of Jiangsu (BK20200244); National Natural Science Foundation of China10.13039/501100001809 (62103324); Open Research Projects of Zhejiang Lab (2019MB0AB02).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.